How to use statistics practice questions

Statistics improves through solving questions, checking the reasoning and correcting the exact step that went wrong. Reading worked examples can help, but it does not show whether you can choose an appropriate method under pressure.

Use each question cycle like this:

  1. Attempt the question without notes. Write down the information given, what is required and any relevant formula.
  2. Show the working. A correct final number is not enough if your method cannot be checked.
  3. Mark the answer carefully. Compare your interpretation, method, arithmetic, units and conclusion.
  4. Record the error. Write a short note such as “used population standard deviation instead of sample standard deviation” or “interpreted correlation as causation”.
  5. Repeat a similar question later. This checks whether the correction has become a usable skill.

Do not complete a large number of questions while repeating the same mistake. A smaller set, analysed properly, usually gives better information about what to revise.

The main types of statistics questions

Most statistics practice questions test one or more of these skills. Your revision set should include all of them rather than only routine calculations.

1. Describing data

These questions ask you to calculate or compare measures such as:

  • mean, median and mode;
  • range, interquartile range and variance;
  • standard deviation;
  • quartiles and percentiles;
  • frequency and relative frequency;
  • features of a table, histogram, box plot or scatter plot.

The important skill is not just calculation. You must also decide which measure describes the data appropriately. For example, the median is often more representative than the mean when a data set contains an extreme value.

2. Probability

Probability questions may involve complements, combined events, conditional probability, independent events, tree diagrams or counting methods. Translate the wording into events before calculating.

Words such as “and”, “or”, “given that” and “at least” have different mathematical meanings. Many errors happen because a student starts calculating before identifying the event being measured.

3. Discrete and continuous distributions

Questions may require you to use a probability distribution, calculate an expected value or find a probability from a model. Check whether the variable is discrete or continuous and whether the probabilities are valid.

For a discrete random variable, the expected value is commonly calculated as:

[ E(X)=\sum xP(X=x) ]

The expected value is a long-run average, not necessarily a result that must occur in one trial.

4. Sampling and data collection

These questions test whether a sample is likely to represent a population. You may need to identify a sampling method, a source of bias, a confounding variable or a weakness in a survey.

A good answer names the problem and explains its effect. “The sample is biased” is less useful than “only volunteers were surveyed, so people with stronger opinions may be over-represented”.

5. Statistical inference

Inference questions can involve confidence intervals, hypothesis tests, significance levels, test statistics, p-values or critical regions, depending on your course.

Keep the structure separate:

  • state the null and alternative hypotheses in context;
  • identify the statistic or probability being used;
  • apply the stated rule or threshold;
  • make a decision;
  • write the conclusion in the language of the question.

Do not treat a p-value as the probability that the null hypothesis is true. It describes how compatible the observed result is with the null model, under the assumptions of the test.

6. Regression and correlation

These questions may ask you to interpret a scatter plot, calculate or interpret a correlation coefficient, use a regression equation or comment on a prediction.

Always check the direction and strength of the relationship, the units of the variables and whether the proposed prediction is inside or outside the observed data range. Correlation alone does not establish that one variable causes the other.

A worked example: summary statistics

Consider the data set:

[ 4,\ 5,\ 5,\ 7,\ 9,\ 10 ]

Find the mean, median and range.

Step 1: Check the order

The values are already in ascending order. This matters for finding the median and range.

Step 2: Calculate the mean

Add the values and divide by the number of observations:

[ \text{mean}=\frac{4+5+5+7+9+10}{6}=\frac{40}{6}=6.67 ]

The mean is approximately 6.67, subject to the required rounding.

Step 3: Calculate the median

There are six values, so the median is the mean of the third and fourth values:

[ \text{median}=\frac{5+7}{2}=6 ]

Step 4: Calculate the range

[ \text{range}=10-4=6 ]

A complete answer includes the method and suitable rounding. If the question asks what the results show, note that the mean is higher than the median because the larger observations pull the mean upwards.

A worked example: probability without guessing the method

A box contains 3 red counters and 2 blue counters. Two counters are selected without replacement. What is the probability that both are red?

The phrase “without replacement” means the composition changes after the first selection. Use a product of conditional probabilities:

[ P(\text{two red})=\frac{3}{5}\times\frac{2}{4}=\frac{6}{20}=\frac{3}{10} ]

The second denominator is 4, not 5, because one counter has already been removed.

A useful checking question is: should the probability of drawing two red counters be smaller than the probability of drawing one red counter? Yes. If your calculation gives a larger value, revisit the setup.

A worked example: interpreting a confidence interval

Suppose a study estimates a population mean and reports a confidence interval from 18.4 to 21.6 units.

A careful interpretation should refer to the method used to create the interval and the population quantity being estimated. For example:

The interval gives a range of plausible values for the population mean, from 18.4 to 21.6 units, at the stated confidence level.

Avoid saying that there is a particular percentage probability that the fixed population mean lies inside this already calculated interval. The precise interpretation depends on the confidence procedure and the assumptions made.

In a practice question, check whether you have:

  • included the units;
  • identified the population mean rather than an individual value;
  • used the confidence level stated in the question;
  • avoided claiming that the interval proves a result is true.

Building a balanced practice set

Create question groups instead of working from one long, mixed list. A useful weekly set might contain:

  • two questions on summarising data;
  • two probability questions, including one conditional problem;
  • one sampling or study-design question;
  • two distribution or expected-value questions;
  • two inference questions;
  • one graph, regression or interpretation question.

Adjust the mix to your syllabus. Check your course specification, textbook or official assessment-body materials for the exact methods and notation required in your examination.

Within each topic, use three levels:

Level 1: routine procedure

These questions give you a familiar data set and ask for a direct calculation. They are useful for learning formulas and calculator procedures.

Level 2: selection and interpretation

These questions require you to choose a method, compare two samples or explain what a result means. They are closer to the decision-making required in an assessment.

Level 3: unfamiliar context

These questions combine several ideas or include incomplete information, a misleading graph or a written claim that must be evaluated. They test whether you understand the statistics rather than recognising a familiar pattern.

Do not move to harder questions only because you can perform the arithmetic. Move on when you can explain why the method is appropriate.

How to mark your own answers

Use an error log with five columns:

Question Error type What I did Correct principle Follow-up
Probability Method Added the two probabilities Events were sequential and without replacement Redo a tree-diagram question

Useful error categories include:

  • Reading: copied a value incorrectly or missed a condition;
  • Method: selected an unsuitable formula or test;
  • Process: set up the calculation incorrectly;
  • Arithmetic: made a calculation or rounding error;
  • Interpretation: gave a conclusion that did not answer the question;
  • Communication: omitted units, hypotheses or context.

After two or three days, attempt one question from each error category without looking at the correction. This is more valuable than simply rereading the answer.

When marking written conclusions, ask: “Could a reader tell what this result means for the original problem?” If not, rewrite it using the variables and context from the question.

Timed practice and calculator checks

Use untimed practice while learning a new method. Then introduce short timed sets once the basic process is reliable. Record the time spent and whether the problem was knowledge, decision-making or arithmetic.

For calculator-based work:

  • write the formula or setup before entering values;
  • keep exact values where possible until the final step;
  • check whether the calculator uses the required statistical mode;
  • compare the output with an estimated size;
  • retain enough working for another person to follow.

A calculator can evaluate a formula, but it cannot decide whether the formula answers the question.

How MySummaries helps

If your revision material is spread across lecture slides, PDFs and handwritten notes, MySummaries can turn your own statistics material into a revision board with flashcards, written mock exams, audio explanations and live oral questioning. Use those generated questions after learning a method, then return to your error log to target the topics you still cannot explain or apply.